Junk: the Fermionic Ansatz

undated (1996-2003 by context) · Notas de física · 649 words · recovered from win/Escritorio/alejo/junk.tex

This is just a remark to . The quotient mapping there can be understood as antisymmetrization with a number of fermion species equal to the dimension of the triple. Our plan is short: to fix notation refreshing the basic issues in Connes argument the argument in then to do our remarks, then fill an additional page with rumiations. See also, for instance, for an easier overview of the junk removal, and for a detailed analysis of the commutative case, which is the one here studied.

Here it is useful to remember that both the clifford algebra and the exterior algebra are defined from the free tensor algebra via a quotient. The free algebra is to be quotiented by the ideal in order to get the corresponding Clifford algebra. On other hand, by using the ideal We quotient the tensor algebra to the exterior one, also called Grassman algebra. In old fashioned index notation, tensors T_ijeⁱe^j are projected to antisymmetric T_ij(eⁱe^j−e^jeⁱ) by removing the symmetrical part, and so with higher tensors, where every pair of consecutive indices is quotiented, then getting a completely antisymmetric tensor.

The coincidence between the number of fermion species and the dimensionality of the space points to a justification of the experimental fact, that we only observe four fermions. This is a conjecture of the author since it surfaced an evening of 1997 at Salerno bay. I have been unable to find any remark on this coincidence, even at speculative publications. A secondary conjecture is that speculative authors are not very used to operate with the tetradic volume form of space-time¹...

While we could prefer a motivation coming from the firm roots of (non-commutative) geometry, links between differential elements and fermionic fields could come from a variety of sources. Directly in field theory, It is known that fermionic fields have no finite implementation in the classical limit; the free Lagrangian being $$-h c \bar\Psi (D + {mc\over h} ) \Psi$$ (, for instance) so only charges and currents of many particles have significance in the h → 0 limit.

To relate fermions with odd differential forms was usual practice in N=2 SUSY, as A. Ibort informed us back in 1992. The algebra of differential forms, has a natural polarization where even forms can be imagined as bosons and odd forms can be seen as fermions, as they commute and anti-commute, respectively. The operators Q₁ = d + d^*, Q₂ = i(d−d^*) play the role of SUSY transformations, and Q₁² = Q₂² = d^*d + dd^* is the Hamiltonian operator.

Last but not least, it is remarkable that the implementation of fermions in the lattice is making heavy use of some implementations of noncommutative differential calculus.